Bibliotheca Polyglotta

If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment.

SI recta linea secta sit vtcunque: Rectangulum sub tota, & vno segmentorum comprehensum, æquale est & illi, quod sub segmentis comprehenditur, rectangulo, & illi, quod a prædicto segmento describitur, quadrato.

一直線。任兩分之。其元線、任偕一分線、矩內直角形與分餘線、偕一分線、矩內直角形。及一分線上直角方形幷等。

For let the straight line AB be cut at random at C; I say that the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC.

For let the square CDEB be described on CB; [I. 46] let ED be drawn through to F, and through A let AF be drawn parallel to either CD or BE. [I. 31] Then AE is equal to AD, CE. Now AE is the rectangle contained by AB, BC, for it is contained by AB, BE, and BE is equal to BC; AD is the rectangle AC, CB, for DC is equal to CB; and DB is the square on CB. Therefore the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC.

Therefore etc. Q. E. D.

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